Some reference in adaptive techniques say that when the solution $u$ is smooth enough we can use p-refinement instead of h-refinement. And when we have for example singularity, we should use h-refinement. I can not understand it. why? When we use a higher order polynomial for approximation we can get smoother solution, so it seems there is no problem for using p-refinement.

• Are you asking for a mathematical justification or some kind of intuitive idea? – knl Sep 17 '17 at 18:41
• No, I just want to know the mathematical justification. – Rosa Sep 17 '17 at 18:43

Limiting considerations to elliptic problems. The approximate solution can converge to exact solution if you do h- or p- adaptivity. If the exact solution is smooth, convergence is faster if you do p- adaptivity. To see this look at a priory error estimator, \begin{equation} \| u - u^h \| \leq C h^p \left\| \frac{\partial^{p+1} u}{\partial x^{p+1}} \right\| \end{equation} where $p$ is polynomial order of base functions. If the solution is smooth, higher order derivatives fast approach zero, so that a priory error estimator. So if you increasing order of base functions, you very quickly converge to an exact solution.
As an example see L-shape body with transport problem,  For details see here http://mofem.eng.gla.ac.uk/mofem/html/mix_transport.html. Mix formulation is applied there, but the same goes for classical finite elements. You can see that both h- and p- adaptivity show convergence, however the fastest convergence you get if you locally at singularity do h- refinement.
• Great answer. But in the formula, the exponent $\mu$ should be $p$, and you need the $(p+1)$th derivative instead of the $p$th. – Wolfgang Bangerth Sep 17 '17 at 22:43